Roll a d20, add modifiers, compare to a DC. There are 4 degrees of success: from equal to the DC to DC+9 is a success, DC+10 or more or a natural 20 are a critical succes. From DC-1 to DC-9 is a failure, DC-10 or lower or a natural 1 are a critical failure. For more details about rounding, multiplying, how to apply modifiers and more, see Checks in the Player Core.

• Move in combat since most things don't have an attack of opportunity (called a "reactive strike").
• Hero points instead of inspiration. You can have up to three, and reset to one at the start of each session, so use them or lose them. A hero point can be spent to reroll, or, spend all your remaining hero points to recover from the dying condition!
• There is no long or short rest. There is only resting. There is also refocussing. Since you don't actually regain many hit points from either activity, you should have health kits and potions to restore health to full between encounters.
• There are very different, but similar rules, for death and dying.
• Ranged damage and spell damage starts out much weaker than melee damage. Only melee attacks and thrown weapons add a modifier to most damage.
• Persistent damage, diseases and poisons often harm you at the end of your turn and then you attempt a save or flat check to cancel them.
• Shields require a free hand and an action each turn. Blocking uses a reaction, only some can do it, and then the shield takes damage. For details see Shields.
• Pathfinder uses a three-action economy. Try to make sure your character uses a third action each turn!
• There are a lot of conditions, and they make a big difference. Adding a condition can be more useful than damage.
• No multiclassing but archetypes are a really neat alternative.
• Invest (attune) up to 10 items per day!
• D&D 3.5e style square counting. Circles aren't squares.

There are several categories of class. Martial vs. Caster vs. Half-caster. Prepared Casters vs. Spontaneous Casters. Divine vs. Primal vs. Arcane vs. Occult Sources. The Rules Lawyer gives some good advice. This is an older video, missing some of the newer classes, but gives a good overview.

Make sure your build has some strong single action abilities to use all of your action economy. Good ones to remember are aid, recall knowledge, demoralize, and step/stride. Aid is always better than a third attack, and often better than a second, because you can help someone crit. Recall knowledge can help you find the enemies' weaknesses to target their lowest defensive stat and possibly trigger extra damage. Fear is one of the most powerful conditions because it affects so many things. Even just walking away from an enemy that can't reactive strike or ranged attack, means that you are trading one of your actions for one of theirs, and the action economy is already tilted in the player's favor.

Because of the four degrees of success, every +1 is at least 5% more chance to succeed and can be 5% more chance to critical succeed, while also reducing chance to fail by 5% and critical fail by 5%. Stacking up various kinds of conditions is important though, since bonuses of the same kind don't stack. Making sure to give lot's of active conditions, of many kinds, with high values, is the key to high level Pathfinder.

Pathfinder has a large number of "activities" that are intended for use during exploration or downtime. Learning some of these could make the non-combat parts of the game more satisfying.

Especially martial characters will want to ensure they meet or exceed the core item progression given here. When you get an extra potency rune, it adds a spot for a property rune. Some of the best property runes for weapons are: crushing, fearsome, rooting, elemental, brilliant, and speed.

• Choose a strong tradition. Many would argue that Arcane > Occult = Nature > Divine in terms of overall power of the tradition. If you have a choice, avoid stacking multiple divine casters in the party.
• Choose strong spells. You have fairly limited spell slots, especially at the beginning, so make sure they can win you the battle. Heal is really strong. It's flexible for many situations and can be scaled up and down well. Sleep and other control spells can be strong, if you manage to take away enemy actions or turns. The action economy generally favors the players and any way you can do an even or better trade of actions you are winning. Spells that give conditions even on a successful save are very powerful.
• Get more spell slots through items so you don't resort to cantrips as often. Potions, Scrolls, Wands and Staves.
• Understand your role. Many times the caster is a kind of controller/buffer/debuffer to maximize the damage of characters with higher damage potential, especially by helping them get better chance to critical hit. Some important ways to do this can be adding a different category of buff or penalty to attack roles than the other characters, using aid, using recall knowledge to discover the weaknesses of the enemy in order to target them, etc.

• Moving for flanking is only the basic positioning help.
• Stepping away from an enemy makes them have to waste an action to move up and close the gap, shifting the action economy in the favor of the party.
• While winning initiative is generally best, if your plan is to debuff a certain enemy with an effect that falls off on the enemy turn such as frightened or sickened, it can be better to delay on your turn until just after the creature goes.
• Rather than spending actions moving up to attack, delaying initiative can also be used to make the enemy burn actions to move to you and saving you from using actions to move, shifting the action economy in the party's favor, while also avoiding putting yourself in a position where you can be surrounded easily. Generally, delay is better than ready for action economy.
• The big hitter should also generally delay until other players go and give buffs and debuffs to maximize the change that they will crit.
• You might be willing to spend actions to ready rather than delay in certain circumstances, such as moving beside an enemy and then readying an attack for when the enemy becomes flanked later. Imagine, a hard hitter delays until after the turn of a friend that will move up to flank, the flanking friend moves and readies a strike for when the hard hitter moves up to flank, then the hard hitter moves up and they both get to attack with flanking, the friend once and the hard hitter potentially twice.

• To use athletics skills, you need to have a free hand. Also, they have the attack tag and affect MAP, so if you want these actions to work you'll need to do them as the first attack on your turn rather than the third. Thus, they are not for the greedy, but make for a great team player setting up your main hitter to do a lot of damage.
• Shoving is awesome. It's like using a move action for your whole party to move away and force the enemy to use an action to close the gap, shifting the action economy in the favor of the party. As part of the shove action, the player could move together with the enemy by one, isolating them from his friends and setting himself up as the only target the enemy can hit without provoking an attack of opportunity. Or maybe you shove an enemy into a flanked position. Or to protect a caster, you can get them away without provoking a reactive strike. Also of note: Shove targets Fortitude rather than AC.
• Tripping is awesome. You make an enemy prone and thus off-guard to everyone, and the enemy needs to waste one of their precious actions to stand up, pushing the action economy in the favor of the party. Combining this with moving away, means that an enemy will need to both stand up and move to close the distance, wasting two of their actions. When a creature stands up, that's a move action that triggers Reactive Strike. Also of note: Trip targets Reflex rather than AC or Fortitude.
• Grapple is awesome. You can control an enemy well. The enemy will have to roll an escape check, losing an action AND incrementing their MAP. And if they fail and have to try again, they are also making the second escape check at -5. Rather than attempt to escape, they will likely strike at the one grabbing them, which could be a good way to tank. Manipulate actions like drawing weapons or casting spells may also fail. And that's just on a normal succes. With a critical succes, the enemy is restrained and can only attempt the escape. Grapple targets Fortitude rather than AC.
• Hiding and concealment vastly improves your chance to avoid being hit, or from the monster's perspective, a greater chance to waste actions, skewing the action economy to the player's favor.